Question

the points ( q(2,-7) ), ( r(-1,-2) ), ( s(-8,2) ), and ( t(-5,-3) ) form a quadrilateral. find the desired slopes and lengths, then fill in the words that best identifies the type of quadrilateral.
answer attempt 1 out of 3
slope of ( overline{qr} = )
length of ( overline{qr} = )
slope of ( overline{rs} = )
length of ( overline{rs} = )
slope of ( overline{st} = )
length of ( overline{st} = )
slope of ( overline{tq} = )
length of ( overline{tq} = )
quadrilateral qrst is

Explanation:

Step1: Calculate slope of $\overline{QR}$

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $Q(2,-7)$ and $R(-1,-2)$, $m_{QR}=\frac{-2-(-7)}{-1 - 2}=\frac{-2 + 7}{-3}=\frac{5}{-3}=-\frac{5}{3}$.

Step2: Calculate length of $\overline{QR}$

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $Q(2,-7)$ and $R(-1,-2)$, $d_{QR}=\sqrt{(-1 - 2)^2+(-2+7)^2}=\sqrt{(-3)^2 + 5^2}=\sqrt{9 + 25}=\sqrt{34}$.

Step3: Calculate slope of $\overline{RS}$

For points $R(-1,-2)$ and $S(-8,2)$, $m_{RS}=\frac{2-(-2)}{-8+1}=\frac{2 + 2}{-7}=-\frac{4}{7}$.

Step4: Calculate length of $\overline{RS}$

For $R(-1,-2)$ and $S(-8,2)$, $d_{RS}=\sqrt{(-8 + 1)^2+(2 + 2)^2}=\sqrt{(-7)^2+4^2}=\sqrt{49+16}=\sqrt{65}$.

Step5: Calculate slope of $\overline{ST}$

For points $S(-8,2)$ and $T(-5,-3)$, $m_{ST}=\frac{-3 - 2}{-5+8}=\frac{-5}{3}=-\frac{5}{3}$.

Step6: Calculate length of $\overline{ST}$

For $S(-8,2)$ and $T(-5,-3)$, $d_{ST}=\sqrt{(-5 + 8)^2+(-3 - 2)^2}=\sqrt{3^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}$.

Step7: Calculate slope of $\overline{TQ}$

For points $T(-5,-3)$ and $Q(2,-7)$, $m_{TQ}=\frac{-7+3}{2 + 5}=\frac{-4}{7}$.

Step8: Calculate length of $\overline{TQ}$

For $T(-5,-3)$ and $Q(2,-7)$, $d_{TQ}=\sqrt{(2 + 5)^2+(-7 + 3)^2}=\sqrt{7^2+(-4)^2}=\sqrt{49+16}=\sqrt{65}$.

Since $m_{QR}=m_{ST}=-\frac{5}{3}$ and $m_{RS}=m_{TQ}=-\frac{4}{7}$ (opposite sides are parallel), and $d_{QR}=d_{ST}=\sqrt{34}$ and $d_{RS}=d_{TQ}=\sqrt{65}$ (opposite sides are equal), the quadrilateral is a parallelogram.

Answer:

slope of $\overline{QR}=-\frac{5}{3}$, length of $\overline{QR}=\sqrt{34}$, slope of $\overline{RS}=-\frac{4}{7}$, length of $\overline{RS}=\sqrt{65}$, slope of $\overline{ST}=-\frac{5}{3}$, length of $\overline{ST}=\sqrt{34}$, slope of $\overline{TQ}=-\frac{4}{7}$, length of $\overline{TQ}=\sqrt{65}$, Quadrilateral $QRST$ is a parallelogram.